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January 4, 2008

On BV Quantization, Part IX: Antibracket and BV-Laplacian

Posted by Urs Schreiber

So far, in my discussion of BV-formalism (part
I,
II,
III,
IV,
V,
VI,
VII, VIII) I had concentrated on the nature and meaning of the underlying complex, without saying a word yet about the antibracket and the BV-Laplacian and the master equation.

I hadn’t mentioned that yet because it wasn’t clear to me yet what the big story here actually is. But now I might be getting closer.

A kind of algebras such that each of them is, essentially, the Clifford algebra of T*X⊕TXT^* X \oplus T X on some spaceXX.

Apparently, this kind of algebra is: BV-algebra.

DefinitionA BV-algebra is a graded commutative algebra AA with an operator Δ:A→A\Delta : A \to A such that Δ2=0\Delta^2 = 0 and such that the “derived bracket” or “antibracket”
[a,b]:=Δ(ab)−Δ(a)b+(−1)|a|aΔ(b)
[a,b] := \Delta(a b) - \Delta(a) b +(-1)^{|a|} a \Delta(b)
is a Gerstenhaber bracket on AA.

The key to seeing this is related to Clifford algebra has been noticed two decades ago in

(This last statement is no secret. The first two statements I haven’t seen emphasized much in the literature.)

From this point of view, one can also see the BV-formalism as a way to conceiving volume forms on spaces XX which are not finite-dimensional manifolds.

The crucial insight which Witten presented in that paper is simply that the BV-Laplacian is nothing but the exterior differential acting non on the standard Clifford irrep build from monomials in Γ(T*X)\Gamma(T^* X), but from the co-standard Clifford irrep build from monomials in Γ(TX)\Gamma(T X).

Hence “antifields” are simply the duals to differential forms, hence are vector fields. (This I have discussed at length before.)

Most crucial seems to be the interpretation of all that:

for toy examples of physics where configuration space is a compact manifold XX, the path integral over that manifold is really the integral over a top-dimensional form which people usually write

e−S/ℏdμ
e^{-S /\hbar} d\mu

but which should best be thought of not as a 0-form times a vomule form, but just as a volume form. To emphasize that, we just
e−S/ℏ
e^{-S /\hbar}
and think of the “dμd\mu” part as absorbed in the exponential.

Then how does that generalize to the standard valilla configuration space in physics, which is generically a space of maps
homS∞(X,Y)
\mathrm{hom}_{S^\infty}(X,Y)
between two manifolds?

To answer that, we need

- a notion of differential forms on generalized smooth spaces

- a notion of “volume forms” for these.

The master equation of BV-formalism is apparently picking these “volume forms” on generalized smooth spaces for us.

Posted at January 4, 2008 5:09 PM UTC

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30 Comments & 1 Trackback

Re: On BV Quantization, Part IX: Antibracket and BV-Laplacian

Take a peek at M. Henneaux “Lectures on the Anti-Field-BRST Formalism for Gauge Theories” Nuclear Physics B (Proceedings Supplement) 18A (1990) 47-106. This explains some of this stuff; in particular see section 8.8. Similarly see Henneaux and Teitelboim “Quantization of Gauge Systems,” Princeton University Press 1992, Theorem 18.1 and section 18.1.3.

And I very much appreciate the clarifications as to the mathematical interpretation of BV formalism which you give there.

But what I haven’t seen you discuss anywhere (but that may well be just my ignorance) is what strikes me as a rather powerful observation, the one by Witten that the above entry is devoted to:

the BV-Laplacian is (best thought of as) nothing but the exterior derivative acting on a not-quite standard Clifford module for Cl(T*X⊕TX)Cl(T^* X \oplus T X).

This I find quite remarkable. I’ll have to re-check again the standard references that “kelly” recommended, which I haven’t actually looked at since long time ago, but this is the kind of good-looking interpretation I was looking for in vain in many reviews of BV-formalism.

In particular, it explains in one stroke not only what the master equation and the antibracket are, but also why we should care!

In most treatments the only reason for “why should we care” that is given is: “look, it works powerful magic”.

Which is fine. But not fully satisfactory.

For instance at the end of your “(secret) homological algebra” you notice that the master equation looks like a Maurer-Cartan equation and then end your article by asking:

But why?

We have talked about this “But why?” before, and maybe I am misinterpreting what “why” you meant, but in Witten’s interpretation of the BV Laplacian which I talked about in my entry above, I find, personally, precisely this “But why?” answered:

Answer: we are really looking at an equation of the form
dω=0
d \omega = 0
with dd the exterior differential, and only because we are looking at this equation not just inside Ω•(X)\Omega^\bullet(X) but inside the larger Cl(T*X⊕TX)\Cl(T^* X \oplus T X) and because we interpret it relative to a slightly nonstandard Clifford module does

- a) the first order differential operator dd superficially look like a second order differential operator Δ\Delta

And on top of all that, it seems that we even begin to see here why this change of perspective is actually useful and important: because it comes from interpreting dd and volume forms on spaces which are not finite dimensional manifolds.

That’s what I like about Witten’s interpretation, which was what the above entry was focusing on.

Re: On BV Quantization, Part IX: Antibracket and BV-Laplacian

The idea is quickly told: the exterior derivative is like the product of a differential operator (partial derivative of functions) with a multiplication operator (wedge product of forms).

Witten points out that the BV-Laplacian is nothing but the odd Fourier transformation of the exterior differential with respect to this odd multiplication operator.

In components, let {xi:ℝn→ℝ}\{x^i : \mathbb{R}^n \to \mathbb{R}\} be the canonical coordinate functions on ℝn\mathbb{R}^n. Then the operator which acts like the exterior derivative on the
Cl(T*ℝn⊕Tℝn)Cl(T^* \mathbb{R}^n \oplus T \mathbb{R}^n)-module Ω•(X)\Omega^\bullet(X)
on ℝn\mathbb{R}^n can be written as

Then on this the exterior derivative acts as the opertor
d=∂∂xi*∂∂xi.
d = \frac{\partial}{\partial x^i{}^*}
\frac{\partial}{\partial x^i}
\,.

Finally one observes that under this transformation the wedge product also undergoes a funny transformation, and we instead end up with the “antibracket”.

I think it all comes down to this:

it looks like for analysis on non-finite dimensional spaces it is possibly not good to consider all differential forms ω\omega as relative to the constant 0-form 1 obtained by wedging
ω=ω∧1
\omega = \omega \wedge 1
but rather to pick something like a volume form μ\mu and regard all differential forms as obtained from that by interior products with “antifields”
ω=ιsomethingμ.
\omega = \iota_{something} \mu
\,.

And apparently the BV-Laplacian/antibracket formalism can be thought of as handling this.

I guess relevant are also the respective remarks which one finds in the second part of Witten’s old Supersymmetry and Morse theory, which point out that the right reference object for forms on loop space is not the constant 0-form, but some “middle form”.

This can actually be traced back also to string theory. One can see that the 0-modes of the supercharges of the type II superstring are nothing but Morse-theoretically deformed exterior derivative and its adjoint on loop space. But the worldsheet vacuum state which they act on is not the constant 0-form on loop space, but some “middle form”.

But the constant 0-form on loop space also plays an important role here: it is the “boundary state” of the space-filling D9-brane.

Re: On BV Quantization, Part IX: Antibracket and BV-Laplacian

A while ago we were talking about Witten’s interpretation of the “BV-Laplace operator” as secretly being nothing but the exterior differential.

I am currently discussing this with Richard Hepworth and Bruce Bartlett in the context of understanding how BRST-BV-quantization is related to Weinstein’s volume forms on stacks (because both address and solve the same kind of problem: to do an integral over an action groupoid by doing it over the underlying space and then dividing out the volume of the gauge group, aka restricting the integral to the gauge orbits).

Anyway, in that context I noticed that in the above entry I never formalized the central statement as I should have. Here it is:

Let XX be a space which we’ll treat as a finite dimensional manifold. Then:

the choice of any volume form on XX, i.e. of a trivialization of the top degree exterior line bundle ω∈∧dim(X)T*X\omega \in \wedge^{\mathrm{dim}(X)} T^* X induces a a degree-reversing isomorphism of graded vector spaces

So anything to be regarded as an equivariant volume form is a closed element in the shifted component of W(g)W(g), in other words: an invariant polynomial on gg:

Volume forms on the L∞L_\infty-algebroid gg are invariant polynomials on gg.

For finite dimensional gg we’d add the condition that the volume form be top dimensional, but for infinite-dimensional field theoretic situations this does not make sense.

So therefore we just speak of any closed form. To reflect that, it becomes useful to shift perspective and describe all elements of W(g)W(g)relative to an unspecified volume form.

So in this sense is the BV complex the shifted cotangent bundle on gg: we mimic the isomorphism
Ω•(X)→≃Tpoly•X
\Omega^\bullet(X)
\stackrel{\simeq}{\to}
T^\bullet_{poly} X
given for XX any finite dimensional manifold by the choice of a volume form and pass hence to the BV-complex being a model for Tpoly•gT^\bullet_{poly} g.

The element 1 in Tpoly•gT^\bullet_{poly} g is now really the volume form, a vector field in there is this volume form contracted with that vector field, etc.

You can go now through large chunks of BV/Schouten/etc.-technology and see that all of this is just nothing but working with Ω•(g)\Omega^\bullet(g) after choosing a volume form and dualizing a bit.

Re: On BV Quantization, Part IX: Antibracket and BV-Laplacian

In this context it is interesting to compare with the theory of integration over supermanifolds. In May I gave a seminar talk on that at HIM in Bonn #, reviewing the standard theory (here are my notes).

This discusses first “volume forms” on supermanifolds, which are not top exterior powers, but certain equivalence classes of choices of bases of the 1-form bundle, where the equivalence relation involves the Berezinian superdeterminant of the change of basis.

(It is probably noteworthy for our discussion of the volume of smooth groupoids/stack # that in this formula the odd-odd part of the change of basis enters with the inverse of its ordinary determinant.)

The main point of interest here is in section 2.5, the definition of integrable forms: given the notioin of volume forms on suoermanifolds, one is interested in those forms generally which may be integrated over super-submanifolds.

Now, an “integrable form” is represented by a pair consisting of a volume form and a (super)polyvector, which we wre to think of as the form obtained by contracting the volume form with the multivector.

So we see. this is precisely the same general idea as that underlying BV-formalism, as discussed above.

Re: On BV Quantization, Part IX: Antibracket and BV-Laplacian

You can go now through large chunks of BV/Schouten/etc.-technology and see that all of this is just nothing but working with Ù •(g) after choosing a volume form and dualizing a bit.

The trouble is I haven’t yet properly understood what the ‘BV formalism’ is all about. This is especially sad since you have written a series of posts entitled “BV for dummies” and I shed a tear each night when I consider the implications of my failure to grok this document :-)

At any event, I understand the following. BV is the central formalism of calculating the path integral in quantum field theory, a mechanism for accounting for the gauge symmetry. Somehow, if we understand BV, we are very close to understanding the path integral.

Now you say that the BV formalism is secretly just another notation for working with the differential forms on the “tangent lie algebroid” of the gauge symmetries.

And the “BV Laplacian”, which I assume is some very important geometric construct in the BV formalism, is secretly the De Rham differential operator in this new notation.

At this point I ask: is your point that we should really be working in the “differential forms” picture all along, and it is confusing to be working in the “polyvector fields” picture?

Or are you saying the opposite?

I didn’t understand your second post about how it relates to integration on supermanifolds.

I would like to be reminded of the answer to the following basic question: what is the outstanding issue of the BV formalism… why are you interested in it? Are you looking for the correct way to understand the BV formalism in terms of the “Lie infinity-theory” picture you have developed? Is it a case of (a) the BV formalism is already well understood, but you would like to see how it fits into your Lie-infinity picture, or (b) there are aspects of the formalism which are not understood, and you believe you can shed light on them using this picture?

Re: On BV Quantization, Part IX: Antibracket and BV-Laplacian

I shed a tear each night when I consider the implications of my failure to grok this document :-)

My fault. Will need to write something clearer. Maybe we can do it together when we have sorted this out.

At this point I ask: is your point that we should really be working in the “differential forms” picture all along, and it is confusing to be working in the “polyvector fields” picture?

My point is just to clarify (for myself) what is really going on. It looks like we should think in terms of differential forms, but work in terms of polyvector fields, which encode differential forms relative to a fixed chosen volume form.

I didn’t understand your second post about how it relates to integration on supermanifolds.

Well, first of all keep in mind that an L∞L_\infty-algebroid can be regarded as a special kind of supermanifold.

When trying to integrate over supermanifolds people long ago ran into the issue that the notion of volume forms as “top exterior powers” does not make sense: the odd 1-forms are even objects and have no finite “top” power.

The solution: define a volume form instead as an equivalence class of local bases of the 1-form bundle, and regard any other “integrable form” (i.e. form which can be integrated over) as a polyvector field, which is to be thought of as giving the desired form after contracting it with the chosen volume form.

So it’s precisely the same kind of idea.

what is the outstanding issue of the BV formalism… why are you interested in it?

The point of BV is to make sense of the idea of “integration over gauge orbits” in the most general situation.

The main theorem in BV says that if SS satisfies the master equation (so that exp(S)exp(S) contracted with the background “volume form” is closed) then the expression of the kind ∫exp(S)δ(gaugecond)\int \exp(S) \delta(gauge cond)
(equation (2.10) here)
is independent of the chosen gauge condition.

So I think BV is all about integrating over ∞\infty-groupoids, or the corresponding L∞L_\infty-algebroids: the space of “paths/trajectories” of some physical system is an L∞L_\infty-groupoid/algebroid and we need to know what it means to integrate over that.

Is it a case of (a) the BV formalism is already well understood, but you would like to see how it fits into your Lie-infinity picture, or (b) there are aspects of the formalism which are not understood, and you believe you can shed light on them using this picture?

Well, maybe a mixture. I suppose you can find plenty of people which say that BV formalism is well understood. But when I look at the literature, even at Costello’s section 2 “Crash course in BV formalism” (page 16) I get the impression that a little more conceptual clarification would be good. But of course maybe that just says that I am confused…

Re: On BV Quantization, Part IX: Antibracket and BV-Laplacian

Urs wrote:

The point of BV is to make sense of the idea of “integration over gauge orbits” in the most general situation.

The integration part I don’t understand but the gauge orbit bit is fine

The BV formalism constructs a differential graded Poisson algebra which is a twisted tensor product of CE(g)CE(g) with a resolution
of something, e.g. the equations of motion. The generators of CE(g)CE(g) aka ghosts are for passing to the quotient.

Ignoring the integrations issue, quantization is supposed to be easy on the BV gadget because it’s free as a CGA
so quantize then reduce homologically rather than trying to quantize after reducing

Re: On BV Quantization, Part IX: Antibracket and BV-Laplacian

I would like to be reminded of the answer to the following basic question: what is the outstanding issue of the BV formalism… why are you interested in it?

I’m interested in it because it’s cohomological physics, that is, all the jazz of antifields, ghosts, etc is easily (for me) seen to be expressible in terms of resolutions in homological algebra.

Are you looking for the correct way to understand the BV formalism in terms of the “Lie infinity-theory” picture you have developed?

I think I’ve already done that. It’s even more transparent in the BFV case. BFV is to BV as Hamiltonian is to Lagrangian.

Is it a case of (a) the BV formalism is already well understood, but you would like to see how it fits into your Lie-infinity picture, or (b) there are aspects of the formalism which are not understood, and you believe you can shed light on them using this picture?

For me, what I don’t understand fully/hardly is how it’s a substitute for path integral quantization.

Re: On BV Quantization, Part IX: Antibracket and BV-Laplacian

Why can we think of any closed form in an infinite-dimensional setting as a volume form?

I don’t think that the claim is that any closed form will do. Instead, the idea is that it’s “middle dimensional forms” which do the trick, i.e. those that involve “every second” coordinate direction in some sense.

I have to admit that I have only a rather vague physicist-style of “understanding” of this. I.e.: no real understanding. There are some remarks along these lines in the second part of Witten’s old article “Supersymmetry and Morse theory”.

It would be good to understand this better. All help is appreciated.

Meanwhile, I read the main insight here as follows: even and in particular if you do not know what exactly the volume form in question is, you can proceed by working with polyvector fields instead which you think of as defining forms once a background volume form is specified, somehow.

Maybe it should be emphasized: BV formalism is supposed to help setting up the right gauge fixed path integral. But the path integral itself is then still a mystery. BV formalism does not help with making the path integral rigorously defined.

But for instance, if you take the path integral to be defined by the sum over renormalized Feynman diagrams, then BV formalism tells you what ghosts and antifields etc. you need to label your diagrams with in order for everything to properly respect the gauge symmetries.

Re: On BV Quantization, Part IX: Antibracket and BV-Laplacian

Urs wrote:

For instance at the end of your `(secret) homological algebra’ you notice that the master equation looks like a Maurer-Cartan equation and then end your article by asking:
But why?
We have talked about this `But why? before, and maybe I am misinterpreting what you meant, but in Witten’s interpretation of the BV Laplacian which I talked about in my entry above, I find, personally, precisely this `But why?’ answered:

Answer: we are really looking at an equation of the form
d??=0
with d the exterior differential, and only because we are looking at this equation not just inside Omega(X) but inside the larger Cl(T *X oplus TX) and because we interpret it relative to a slightly nonstandard Clifford module does
- a) the first order differential operator d superficially look like a second order differential operator
- b) does the linear d turn into the non-linear \Delta S+{S,S}.

Jim: My `but why?’ was meant at a deeper level,
so let me ramble:

First of all, I misspoke when I said the master eqn terminology was becoming dominant; in fact, it’s the Maurer-Cartan or MC that’s now dominant. I take it that this is entirely on the basis of the form of the eqn, the original MC being for diff forms on a Lie group.

So at the top level: It holds for Lie groups G because TG is trivializeable
(G is parallelizeable).

Historically I’d bet that regarding it as the equation of flatness came earlier, but doesn’t provide a catchier title.

Relation to deformation theory expessed in terms of a differential d s/t d^2=0.
A deformation can be described in terms of
d+p s/t (d+p)^2 =0 which can be expanded
to [d,p]+p^2=0 since d^2=0. In char \neq 2, p^2 = 1/2 [p,p] et voila!

A particular case of that occurs for twisting cochains p - regarding p (called: perturbation) as giving a deformation of d the differential on B x F so that the differential d+p corresponds to a fibration F –> E –> B.

The distinction between geometry and homotopy theory shows up in that finding a flat connection is not always possible for a given bundle but a twisting cochain always exists.

That MC appears as the Master Equation I take it as reflecting the fact that a bundle (at least the corresponding qDGCA)
is in hte background. The quantum ME is
then related to regarding quantization as deformation.

Re: On BV Quantization, Part IX: Antibracket and BV-Laplacian

Urs wrote:

it seems that we even begin to see here why this change of perspective is actually useful and important: because it comes from interpreting d and volume forms on spaces which are not finite dimensional manifolds.

That’s what I like about Witten’s interpretation, which was what the above entry was focusing on.

JIM: Much (all?) of that can be consider as formal, cf. Gel’fand’s formal differential geometry. But that would avoid having to discuss smootheness!! Why not?

Re: ME: the Master Equation

Once upon a time,
dw=±[w,w]
d w = \pm [w,w]
was note as flatness
In deformation theory, aka integrability
In homotopy theory, twisting cochain

Now some powerful lobby (derived from the BV camp, I believe) has called it

The Master Equation

and the world seem to be follwoing suit.

Yes, this is the attitude I am trying to follow here. I don’t like thinking of weird sounding “master equations”. Instead, I am thinking that what people find to be very important but address by funny names should actually be something essentially standard and well known appearing to us in slightly different guise.

So I want to understand: why is the flatness condition called the “master equation” the key to perturbative quantization of gauge systems?

There must be a good powerful general interpretation.

Possibly the answer is well known (maybe to you, too, but then I haven’t absorbed it from you yet, sorry) to somebody. But certainly I am not the only one puzzled about this (even though not everybody seems to express his or her puzzlement). So in any case I have the feeling it is justified to talk more about this stuff.

Witten’s anti-bracket formalism

\begin{document}
Thanks to Urs for forcing me to go back and reread Witteen’s ` A note on the anti-bracket formalism’.
I had long since appreciated part of it and apparently forgotten what I did not apprreciate,
especially the Clifford part.

Much of the following is a translation of Witten into my language, as I think I saw it at the time.
The quadratic form in question is just a dual pairing - in this case, of tangent and cotangent vectors
on a *finite dimensional* manifold M.
Extending this to the graded comm alg they generate gives the anti-bracket, though that’s not how Witten sees it. From my point of view, $TM x_M T*M$ is a graded symplectic manifold.

Then Witten considers irreducible reps of this graded Lie algebra. To do this locally, he uses
the free GC algebras, one generated by $ dx^i$ and and the other by $\partial_x^j$. now comes the
`physics’: treat in the first case $ dx^i$ as creation operators and $\partial_x^j$ as annihilators,
while for the second case, vice versa (or in phys notation ). That is, interchange the roles of the generators. Notice that we now have representations as $\partial_{dx^i}$ and $\partial_{\partial_{x^j}}.$

Thus, depending on what we consider as the variable, these are either first order or second order,
which Witten keeps track of by pseudonysms: $z^i$ and $w_j$.

It is in establishing an isomorphism (NOT unique) that volume form appears.

He then recovers the anti-bracket (I consider it to be there already) as the deviation of $Delta$ from being a derivation - cf. Gerstenhaber.

Witten finds it `is striking’ that infinitesimal transformations have the form of gauge invariance in OSFT,
but later work has shown this is generic for Lagrangian variations.

He calls for `a Lagrangian whose variational equation would be the qME (quantum master equation)’.
I have a feeling this has been done but can’t surface a reference.

Re: Witten’s anti-bracket formalism

Thanks a lot, Jim, for your comments! One remark: you write:

what I did not apprreciate,
especially the Clifford part.

[…]

From my point of view, TMxMT*MTM x_M T*M is a graded symplectic manifold.

Okay, good. So maybe that clarifies much of what we had been discussing at various points, latey: I kept saying that it might be worthwhile to think of a symplectic dg-manifold (what people call symplectic nn-algebroids) as a kind of Clifford structure.

Anyway, have to call it quits now. More later. Thanks again for your comments.

Re: Witten’s anti-bracket formalism

Thinking of it as Clifford must ‘mean’ something to you or your intuitiion - but what?

One way to put it is this:

we know that every graded-commutative dga comes to us

i) as the CE-algebra of some semistrict Lie ∞\infty-algebra

ii)
as the dga of differential forms on some generalized space (on some presheaf).

If I know what Ω•(X)\Omega^\bullet(X) means for a generalized smooth space, there is nothing more natural than trying to define what Cl(T*X⊕TX)Cl(T^* X \oplus T X) means for a general smooth space. It is something we should want to look at anyway.

Re: On BV Quantization, Part IX…

It might be helpful for anyone who wants to reproduce the above algebra to point out that Witten’s seminal paper Mod.Phys.Lett.A5 (1990) 487 contains two misprints, which have proliferated into this blog at various places. A preprint version of Witten’s paper can be downloaded from the KEK preprint library:

To approach a path integral, we might want to consider such pulled back metric on all plots U×IU \times I, then do the fiber integration just over II – this should yield the kinetic part of the actions. The remaining part of the volume element would be the path integral measure – on that finite dimensional plot.

Then the whole subtlety is in how to, somehow, take the colimit over all plots, or something.

I am thinking: we want for plots of all dimensions a volume form. So maybe we should consider something like
exp(gijdxi⊗dxj)\exp( g_{ij} d x^i \otimes d x^j ) with the pulled back metric

(written here in components to emphasize the tensor product structure) and then form the exponential in wedge products.

Then doing a kind of fiber integration over the [0,1][0,1]-factor would, by the above observation, actually produce the kinetic action part. And even exponentiated.

Phew, I am getting too tired. I’ll say all this again more coherently and more nicely tomorrow.

But if there is anyone out there who has already heard and/or thought about integrals over diffeological spaces – please drop me a note.

Re: Integration over Diffeological Spaces

I find this seems interesting but very difficult to understand. Basically what I am getting is that you are saying: “Making sense of integrating things over diffeological spaces is the same as making sense of the path integral.” That seems clear. But then you take it a bit further and talk about fiber integration over II and the kinetic action parts and then I am lost.

Read the post Frobenius Algebras and the BV FormalismWeblog: The n-Category CaféExcerpt: Bruce Bartlett is looking at the latest article by Cattaneo and Mnev on BV-quantization of Chern-Simons theory.Tracked: November 17, 2008 5:00 PM